# nLab bubble of nothing

Contents

### Context

#### Gravity

gravity, supergravity

## Phenomenology

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

A bubble of nothing in the sense of Witten 82 is an instability of the Kaluza-Klein vacuum where a gravitational instanton mediates the decay of a $5$-dimensional Euclidean Schwarzschild spacetime into nothing: the hole in space grows at the speed of light until it hits null infinity?, leading to the “annihilation” of spacetime.

## Construction of the solution

Let us start from the spacetime $(\mathbb{R}^{4}-B_R^3)\times S^1$, where $B_R^3$ is a ball of radius $R$ and $S^1$ is a circle of radius $R_5$, equipped with the following Euclidean Schwarzschild metric:

$g \;=\; r^2\mathrm{vol}_{S^3} + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2,$

where $\theta$ is the coordinate on the circle and $r$ is the radius of $\mathbb{R}^{4}$.

Let the volume of the $3$-sphere be $\mathrm{vol}_{S^3} = \mathrm{d}\chi^2 + \sin(\chi)\mathrm{vol}_{S^2}$. We can now go back to the Minkowski signature by the Wick rotation $\psi := -i\chi + i\frac{\pi}{2}$. Thus, we have

$g \;=\; -r^2\mathrm{d}\psi^2 + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + r^2\cosh^2(\psi)\mathrm{vol}_{S^2} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2.$

## Properties

By change of coordinates $\rho := r\cosh(\psi)$ and $t:=r\sinh(\psi)$, we see that at large radius the metric goes to the Minkowski metric

$\lim_{r\rightarrow \infty}g \;=\; -\mathrm{d}t^2 +\mathrm{d}\rho^2 + \rho^2\mathrm{vol}_{S^2}+R_5^2\mathrm{d}\theta^2.$

However, the coordinates $(t,\rho)$ parametrize all the $4$-dimensional base manifold but the region $\{ \rho^2 - t^2 \leq R \}$ of the bubble. The boundary of the bubble in the $4$-dimensional base space has a radius which grows with time as

$\rho_{\mathrm{BON}}(t) \;=\;\sqrt{R^2+t^2}.$

Moreover, the effective size of the circle compactification will be depend on $r$ by

$R_{5}(r) \;=\; R_5\sqrt{1-\frac{R^2}{r^2}},$

so that it goes to $0$ for $r\rightarrow R$ and it goes to $R_5$ for for $r\rightarrow \infty$.

## References

Last revised on April 21, 2021 at 11:48:41. See the history of this page for a list of all contributions to it.